We can rewrite the equality as:
(1−v2)±u1−u21−v2p=v(1−u2)(1-v^2)\pm u\sqrt{1-u^2}\sqrt{1-v^2}p=v(1-u^2)(1−v2)±u1−u21−v2p=v(1−u2)
The sign is chosen depending on the sign of expression cos x cos ycos\,x\,cos\,ycosxcosy .
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